In order to increase the performance of a compressor or other rotating machine, it is often desirable to increase the length of the shaft supporting the rotating elements of the machine. As the length is increased, the critical rotation frequencies at which vibrations take place are reduced. The vibrations are related to the natural sag between the ends of the shaft supported by bearings at each end. While placing additional bearings near the center of the shaft has been considered, it is a considerable drawback. Access to the bearing is very difficult, if not impossible, without disassembling the entire machine. One proposal is to use magnetic bearings near the midpoint between the ends of the shaft as explained in U.S. Pat. No. 4,141,604. It is an object of the present invention to damp the vibrations of a rotating shaft and to increase the critical frequency without requiring addition of inaccessible bearings.